In many applications, data are of high dimension. How to analyze all the dimensions simultaneously for various applications such as personalized recommendation and summarization is a challenging issue.
Pairwise relationship exists in many applications and dyadic data analysis has been studied extensively by various researchers. For example, the well known Latent Semantic Indexing (LSI) focuses on dyadic data consisting of term-document pairs. However, in many applications, data are polyadic, i.e., they are of higher dimensions. Networked data are such an example: in a research paper corpus such as CiteSeer, an author, in an article on a specific topic, cites a reference. In this example, a data record is an author-topic-reference triple, i.e., with dimension of three.
Collaboratively tagging data is another example: in collaborating tagging systems such as Del.icio.us, a user assigns a set of tags to a given url (which corresponds to a Web page). Here data records are user-tag-url triples. Combining all aspects of the data into data analysis is a challenging issue and various approaches have been proposed for fusing information into a single framework. Most existing research work only analyzes pairwise relationship among different dimensions, and then combines the analysis results afterwards. Such an approach loses the higher order (higher than pairwise) dependency among various dimensions of data.
Some studies have used a set of concepts to capture all the pairwise relationship simultaneously. Because these approaches use the same concepts to represent all the pairwise relationship among various dimensions, they offer better performance than those approaches that consider the pairwise relationships independently. This second approach often gives better performance because it describes the real data using a more accurate model. However, this approach usually uses a linear combination to fuse all pairwise relationships. Such a linear combination is somewhat ad hoc—it is difficult to find a good intuition behind the coefficients as well as principled ways to set the values of the coefficients.